B2. MODELING CORE Directors M. Covert and K. C. Huang; supporting participation by J. Ferrell and C. Tomlin Ariel Jaimovich (Postdoc). Dr. Jaimovich is a systems engineer between Pat Brown and Tobias Meyer's laboratory who will provide advise on modeling approaches and also help in coordinating the summer outreach workshop of the center. The overarching aim of the Modeling Core is to use models of cellular systems as a tool to investigate the genetic and molecular networks that underpin the collective responses of cells. These insights can then be tested experimentally using chemical, light, or mechanical perturbations, and then assessing the consequences, whenever possible, through real-time, single cell microscopy and quantitative image analysis. Both the dynamical and quantitative natures of this type of experimental approach are of critical importance. The Center's faculty members have extensive experience and success in modeling the responses of individual cells, and in integrating these models with quantitative experimentation. The next challenge is to couple single cell models together to allow the collective behaviors of cells to be analyzed and understood. At the single cell level there are many possible approaches to modeling, ranging from course-grained methods like Boolean modeling through detailed master equation-based stochastic modeling [121]. Arguably the most successful approaches have been those based chemical kinetic theory, particularly ordinary differential equation (ODE) models. ODE models rely upon the assumption that cells, or at least specific compartments within cells, act like well-stirred systems; when this assumption is in doubt, ODE models should be considered at best a first step toward an understanding of the system. Challenges in ODE modeling include deciding how detailed the molecular description is to be. For example, the cell cycle regulator APC/C is phosphorylated at least 71 sites [122], and so a detailed ODE description of APC/C regulation could include 2nd distinct phosphoisomers, an impractically large number. On the other hand, quantitative studies of the response of APC/C to CDK1 have shown that it behaves like a digital switch (Yang and Ferrell, unpublished), which provides a rationale for approximating this complex regulation as a two-state system. Sometimes the overall time lags produced in complex processes, like transcription, are more important than the details of the intermediate steps of the process. In these cases delay differential equation models (DDEs) may be appropriate, although the discrete time lags assumed in DDEs can lead to unrealistic modeled behaviors. We have made use of DDE models, detailed ODE models, and simplified ODE models designed to capture the essence of a process. Each type of modeling has can be useful. The next level of complexity in single cell modeling is to incorporate spatial considerations. Sometimes compartmentalized ODEs are sufficient for this purpose. For example, in the studies of spatial positive feedback in the mitotic trigger recently published by the Ferrell and Meyer labs, we made use of compartmentalized ODEs because it seemed likely that the time scale for diffusive mixing of cyclin B1-CDK1 within the cytoplasm or nucleus (seconds) was substantially faster than the time scale for equilibration between the cytoplasm and nucleus (minutes) [29]. At other times, considering a full continuum of spatial states is essential and PDE modeling is appropriate. For these reasons, PDE modeling was used in the analysis of mitotic trigger waves described in Aim 1.1. Once satisfactory single cell models have been obtained, the next level of complexity is to couple these models to allow collective cellular behaviors to be modeled and analyzed: When small numbers of cells in defined geometries are being considered-for example, in analyzing early Xenopus embryos-coupled ODEs are feasible. In other cases, such as the modeling carried out by the Axelrod and Tomlin labs on differentiation in epithelial sheets, or for processes involving both biochemical and mechanical interactions, hybrid models may be more appropriate. Discrete and agent-based modeling, where each interaction changes an output vector of important parameters like location, strain, velocity, or secretion, provide another way of approaching the collective behavior of cells. All of these modeling approaches share a number of challenges. These include uncertainties in the representation of individual cells using ODEs and PDEs. Even in well-studied processes, like cell cycle regulation, the network topology of system is likely incomplete and parts might be incorrect. Model parameter values are often not well defined. However, estimation of parameters is helped by understanding the underlying biochemical mechanisms, which emphasizes the importance of close interaction between quantitative biologists and modelers. It is also challenging to introduce stochasticity and noise into a model in the best way. Approaches include direct Monte Cario simulations and the inclusion into ODE models of noise functions. In any case, all models yield a mix of high and low confidence predictions that need to be identified and represented/visualized.